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3. Activity Coefficients of Aqueous Species 3.1. Introduction

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3. Activity Coefficients of Aqueous Species 3.1. Introduction 3. Activity Coefficients of Aqueous Species 3.1. Introduction The thermodynamic activities (ai) of aqueous solute species are usually defined on the basis of molalities. Thus, they can be described by the product of their molal concentrations (mi) and their γ molal activity coefficients ( i): γ ai = mi i (77) The thermodynamic activity of the water (aw) is always defined on a mole fraction basis. Thus, it can be described analogously by product of the mole fraction of water (xw) and its mole fraction λ activity coefficient ( w): λ aw = xw w (78) It is also possible to describe the thermodynamic activities of aqueous solutes on a mole fraction (x) basis. However, such mole fraction-based activities (ai ) are not the same as the more familiar (m) molality-based activities (ai ), as they are defined with respect to different choices of standard λ states. Mole fraction based activities and activity coefficients ( i), are occasionally applied to aqueous nonelectrolyte species, such as ethanol in water. In geochemistry, the aqueous solutions of interest almost always contain electrolytes, so mole-fraction based activities and activity co- efficients of solute species are little more than theoretical curiosities. In EQ3/6, only molality- (m) based activities and activity coefficients are used for such species, so ai always implies ai . Be- cause of the nature of molality, it is not possible to define the activity and activity coefficient of (x) water on a molal basis; thus, aw always means aw . Solution thermodynamics is a construct designed to approximate reality in terms of deviations from some defined ideal behavior. The complex dependency of the activities on solution compo- sition is thus dealt with by shifting the problem to one of describing the activity coefficients. The usual treatment of aqueous solutions is one which simultaneously employs quantities derived from, and therefore belonging to, two distinct models of ideality (Wolery, 1990). All solute ac- tivity coefficients are based on molality and have unit value in the corresponding model of ide- ality, called molality-based ideality. The activity and activity coefficient of water are not constant in an ideal solution of this type, though they do approach unit value at infinite dilution. These solvent properties are derived from mole fraction-based ideality, in which the mole fraction ac- tivity coefficients of all species components in solution have unit value. In an ideal solution of this type, the molal activity coefficients of the solutes are not unity, though they approach it at infinite dilution (see Wolery, 1990). Any geochemical modeling code which treats aqueous solutions must provide one or more mod- els by which to compute the activity coefficients of the solute species and the solvent. In many γ codes, what is computed is the set of i plus aw. As many of the older such codes were constructed to deal only with dilute solutions in which the activity of water is no less than about 0.98, some of these just take the activity of water to be unity. With the advent of activity coefficient models - 36 - of practical usage in concentrated solutions (mostly based on Pitzer’s 1973, 1975 equations), there has been a movement away from this particular and severe approximation. Nevertheless, it is generally the activity of water, rather than the activity coefficient of water, which is evaluated from the model equations. This is what was previously done in EQ3/6. However, EQ3/6 now γ λ evaluates the set of i plus w. This is done to avoid possible computational singularities that may arise, for example if heterogeneous equilibria happen to fix the activity of water (e.g., when a solution is saturated with both gypsum and anhydrite). Good models for activity coefficients must be accurate. A prerequisite for general accuracy is thermodynamic consistency. The activity coefficient of each aqueous species is not independent of that of any of the others. Each is related to a corresponding partial derivative of the excess Gibbs energy of the solution (GEX). The excess Gibbs energy is the difference between the com- plete Gibbs energy and the ideal Gibbs energy. Because there are two models of ideality, hence two models for the ideal Gibbs energy, there are two forms of the excess Gibbs energy, GEXm (molality-based) and GEXx (mole fraction-based). The consequences of this are discussed by Wolery (1990). In version 7.0 of EQ3/6, all activity coefficient models are based on ideality de- fined in terms of molality. Thus, the excess Gibbs energy of concern is GEXm. The activity of wa- ter, which is based on mole-fraction ideality, is imported into this structure as discussed by Wolery (1990). The relevant differential equations are: EXm 1 ∂ γ ------------------------G ln i = ∂ (79) RT ni EXm Σ 1 ∂ -------m- ------------------------G ln aw = – Ω + ∂ (80) RT nw where R is the gas constant, T the absolute temperature, Ω the number of moles of solvent water comprising a mass of 1 kg (Ω ≈ 55.51),and: Σ mm= ∑ i (81) i the sum of molalities of all solute species. Given an expression for the excess Gibbs energy, such equations give a guaranteed route to thermodynamically consistent results (Pitzer, 1984; Wolery, 1990). Equations that are derived by other routes may be tested for consistency using other rela- tions, such as the following forms of the cross-differentiation rule (Wolery, 1990): ∂ ln γ ∂ lnγ --------------j -------------i ∂ = ∂ (82) mi mj ∂ ln a ∂ ln γ ----------------w --------------i ------1 ∂ = ∂ – (83) ni nw nw - 37 - In general, such equations are most easily used to prove that a set of model equations is not ther- modynamically consistent. The issue of sufficiency in proving consistency using these and relat- ed equations (Gibbs-Duhem equations and sum rules) is addressed by Wolery (1990). The activity coefficients in reality are complex functions of the composition of the aqueous so- lution. In electrolyte solutions, the activity coefficients are influenced mainly by electrical inter- actions. Much of their behavior can be correlated in terms of the ionic strength, defined by: 1 2 I = ---∑m z (84) 2 i i i where the summation is over all aqueous solute species and zi is the electrical charge. However, the use of the ionic strength as a means of correlating and predicting activity coefficients has been taken to unrealistic extremes (e.g., in the mean salt method of Garrels and Christ, 1965, p. 58- 60). In general, model equations which express the dependence of activity coefficients on solu- tion composition only in terms of the ionic strength are restricted in applicability to dilute solu- tions. The three basic options for computing the activity coefficients of aqueous species in EQ3/6 are models based respectively on the Davies (1962) equation, the “B-dot” equation of Helgeson (1969), and Pitzer’s (1973, 1975, 1979, 1987) equations. The first two models, owing to limita- tions on accuracy, are only useful in dilute solutions (up to ionic strengths of 1 molal at most). The third basic model is useful in highly concentrated as well as dilute solutions, but is limited in terms of the components that can be treated. With regard to temperature and pressure dependence, all of the following models are parameter- ized along the 1 atm/steam saturation curve. This corresponds to the way in which the tempera- ture and pressure dependence of standard state thermodynamic data are also presently treated in the software. The pressure is thus a function of the temperature rather than an independent vari- able, being fixed at 1.013 bar from 0-100°C and the pressure for steam/liquid water equilibrium from 100-300°C. However, some of the data files have more limited temperature ranges. 3.2. The Davies Equation The first activity coefficient model in EQ3/6 is based on the Davies (1962) equation: γ 2I log= – Aγ, z --------------- + 0.2I (85) i 10 i 1 + I (the constant 0.2 is sometimes also taken as 0.3). This is a simple extended Debye-Hückel model (it reduces to a simple Debye-Hückel model if the “0.2I” part is removed). The Davies equation is frequently used in geochemical modeling (e.g., Parkhurst, Plummer, and Thorstenson, 1980; Stumm and Morgan, 1981). Note that it expresses all dependence on the solution composition through the ionic strength. Also, the activity coefficient is given in terms of the base ten loga- rithm, instead of the natural logarithm. The Debye-Hückel Aγ parameter bears the additional label “10” to ensure consistency with this. The Davies equation is normally only used for temperatures close to 25°C. It is only accurate up to ionic strengths of a few tenths molal in most solutions. In - 38 - some solutions, inaccuracy, defined as the condition of model results differing from experimental measurements by more than the experimental error, is apparent at even lower concentrations. In EQ3/6, the Davies equation option is selected by setting the option flag iopg1 = -1. A support- ing data file consistent with the use of a simple extended Debye-Hückel model must also be sup- plied (e.g., data1 = data1.com, data1.sup, or data1.nea). If iopg1 = -1 and the supporting data file is not of the appropriate type, the software terminates with an error message. The Davies equation has one great strength: the only species-specific parameter required is the electrical charge. This equation may therefore readily be applied to a wide spectrum of species, both those whose existence is well-established and those whose existence is only hypothetical. The Davies equation predicts a unit activity coefficient for all neutral solute species. This is known to be inaccurate. In general, the activity coefficients of neutral species that are non-polar (such as O2(aq), H2(aq), and N2(aq)) increase with increasing ionic strength (the “salting out ef- fect,” so named in reference to the corresponding decreasing solubilities of such species as the salt concentration is increased; cf.
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